Geometric Manifestations of Positivity
This chapter focuses on a number of results that in one way or another express geometric consequences of positivity. In the first section we prove the Lefschetz hyperplane theorem following the Morse-theoretic approach of Andreotti-Frankel . Section 3.2 deals with subvarieties of small codimension in projective space: we prove Barth’s theorem and give an introduction to the conjectures of Hartshorne. The connectedness theorems of Bertini and Fulton-Hansen are established in Section 3.3, while applications of the Fulton-Hansen theorem occupy Section 3.4. Like the results from 3.2, these reflect the positivity of projective space itself. Finally, some extensions and variants are presented in Section 3.5.
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